C13034 - Physics Around Mirror Symmetry

G_2 manifolds play the analogous role in M-theory that Calabi-Yau manifolds play in string theory. There has been work in the physics community on conjectural "mirror symmetry" in this context, and it has also been observed that singularities are necessary for a satisfactory theory. After a very brief review of these physical developments (by a mathematician who doesn't necessarily understand the physics), I will give a mathematical introduction to G_2 conifolds. I will then proceed to give a detailed survey of recent mathematical developments on G_2 conifolds, including desingularization, deformation theory, and possible constructions of G_2 conifolds. This includes separate joint works of myself with Jason Lotay and with Dominic Joyce.

Chiral gauge theories in two dimensions with (0,2) supersymmetry admit a much broader, and more interesting, class of vacuum solutions than their better studied (2,2) counterparts. In this talk, we will explore some of the possibilities that are offered by this additional freedom by including field-dependent theta-angles and FI parameters. The moduli spaces that will result from this procedure correspond to heterotic string backgrounds with non-trivial H-flux and NS-brane sources. Along the way, a remarkable relationship between (0,2) gauge anomalies and H-flux will emerge.

Topological string theory is restricted enough to be solved completely in the perturbative sector, yet it is able to compute amplitudes in physical string theory and it also enjoys large N dualities. These gauge theory duals, sometimes in the form of matrix models, can be solved past perturbation theory by plugging transseries ansätze into the so called string equation. Based on the mathematics of resurgence, developed in the 80's by J. Ecalle, this approach has been recently applied with tremendous success to matrix models and their double scaling limits (Painlevé I, etc). A natural question is if something similar can be done directly in the topological closed string sector. In this seminar I will show how the holomorphic anomaly equations of BCOV provide the starting point to derive a master equation which can be solved with a transseries ansatz. I will review the perturbative sector of the solutions, its structure, and how it generalizes for higher instanton nonperturbative sectors. Resurgence, in the guise of large order behavior of the perturbative sector, will be used to derive the holomorphicity of the instanton actions that control the asymptotics of the perturbative sector, and also to fix the holomorphic ambiguities in some cases. The example of local CP^2 will be used to illustrate these results. This work is based on 1308.1695 and on-going research in collaboration with J.D. Edelstein, R. Schiappa and M. Vonk.

I will describe a new method for understanding a large class of generalized complex manifolds, in which we view them as usual symplectic structures on a manifold with a kind of log structure. I will explain this structure in detail and explain how it can be used to prove a Tian-Todorov unobstructedness theorem as well as topological obstructions for existence of nondegenerate generalized complex structures.

The concept of wall-crossing structure (WCS for short) was introduced recently in my joint work with Maxim Kontsevich. WCS appear in different disguises in the theory of Donaldson-Thomas invariants of Calabi-Yau 3-folds, quiver representations,integrable systems of Hitchin type, cluster algebras, Mirror Symmetry, etc.I plan to discuss the definition of WCS and illustrate it in several well-known examples. If time permits I will speak about a special class of WCS called rational WCS. It gives rise to wall-crossing formulas with factors which are algebraic functions. Conjecturally such WCS appear in Hitchin integrable systems with singularities.

Hermitian matrix models have been used since the early days of 2d quantum gravity, as generating series of discrete surfaces, and sometimes toy models for string theory. The single trace matrix models (with measure dM exp( - N Tr V(M)) have been solved in a 1/N expansion in the 90s by the moment method of Ambjorn et al. Later, Eynard showed that it can be rewritten more intrinsically in terms of algebraic geometry of the spectral curve, and formulated the so-called topological recursion. In a similar way, we will show that double hermitian matrix models are solved by the same topological recursion, and more generally, that arbitrary hermitian matrix models are solved by a "blobbed topological recursion", whose properties still have to be investigated.

I will review some ongoing work on the low energy properties of D-branes/boundary conditions in massive two-dimensional field theories with (2,2) supersymmetry.

We compute the partition function on the hemisphere of a class of two-dimensional (2,2) supersymmetric field theories including gauged linear sigma models. The result provides a general exact formula for the central charge of the D-brane placed at the boundary. It takes the form of Mellin-Barnes integral and the question of its convergence leads to the grade restriction rule concerning branes near the phase boundaries. We find expressions in various phases including the large volume formula in which a characteristic class called the Gamma class shows up. The two sphere partition function factorizes into two hemispheres glued by inverse to the annulus. The result can also be written in a form familiar in mirror symmetry, and suggests a way to find explicit mirror correspondence between branes.

We apply supersymmetric localization to N=(2,2) gauged linear sigma models on a hemisphere, with boundary conditions, i.e., D-branes, preserving B-type supersymmetries. We explain how to compute the hemisphere partition function for each object in the derived category of equivariant coherent sheaves, and argue that it depends only on its K theory class. The hemisphere partition function computes exactly the central charge of the D-brane, completing the well-known formula obtained by an anomaly inflow argument. We also formulate supersymmetric domain walls as D-branes in the product of two theories.  We exhibit domain walls that realize the sl(2) affine Hecke algebra.  Based on arXiv:1308.2217.

I will discuss a class of limiting points in the moduli space of d=2 (2,2) superconformal field theories.  These SCFTs arise as IR limits of "hybrid" UV theories constructed as a fibration of a Landau-Ginzburg theory over a base Kaehler geometry.  A significant generalization of Landau-Ginzburg and large radius geometric limit points, the hybrid theories can be used to probe general features of (2,2) and (0,2) SCFT moduli spaces.

Mathieu Moonshine is a striking and unexpected relationship between the sporadic simple finite group M24 and a special Jacobi form, the elliptic genus, which arises naturally in studies of nonlinear sigma models with K3 target.  In this talk, we first discuss its predecessor (Monstrous Moonshine), then discuss the current evidence in favor of Mathieu Moonshine.  We also discuss extensions of this story involving `second quantized mirror symmetry,' relating heterotic strings on K3 to type II strings on Calabi-Yau threefolds.